On a national mathematics assessment test, a random sample of 120 twelfth grade

Question

On a national mathematics assessment test, a random sample of 120 twelfth grade students has a mean score of 150 with a standard deviation of 34. This test result prompts a school administrator to claim that the mean score for all twelfth grade students who took the test is at least 145. At alpha = 0.05, can you reject the administrator's claim? a. Give H_0 and H_a and state which one is the claim. b. Sketch the normal distribution, label the critical value(s), and shade the rejection region. c. Find the standardized test statistic and tell whether to reject or fail to reject H_0. d. Interpret the decision in the context of the original claim.

Explanation / Answer

Solution:-

a) State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: > 145

Alternative hypothesis: < 145

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a one-sample t-test.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = s / sqrt(n)

S.E = 3.104

DF = n - 1 = 120 - 1

D.F = 119

t = (x - ) / SE

c) t = 1.61

tcritical = 1.658

Critical region is t greater than 1.658.

where s is the standard deviation of the sample, x is the sample mean, is the hypothesized population mean, and n is the sample size.

The observed sample mean produced a t statistic test statistic of 1.61. We use the t Distribution Calculator to find P(t > 1.61) = 0.055

Thus the P-value in this analysis is 0.055

Interpret results. Since the P-value (0.055) is greater than the significance level (0.01), we cannot reject the null hypothesis.

d) From the above test we have sufficient evidence in the favor of the claim that mean score for all twelfth grade students who took the test is atleast 145.

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